I was fascinated by the function exp^(1/2)[ln^(1/2)[x]+1] - x where x is a real > 0.

This function must be smaller than x.

But im puzzled by its growth rate on the real line and its singularity growth rate on the complex plane.

Maybe you guys know more.

Regards

tommy1729

To clarify this a bit more.

We know that this function f(x) = exp^(1/2)[ln^(1/2)[x]+1] - x must be smaller than x and greater than 0 and we know that if we iterate f(x)+x 'x' times we get (about) exp^(1/2)[x] which grows faster than any polynomial and slower than any exponential ( any base ).

( exp^(1/2) is like a superfunction here )

An asymptotic expression for f(x) in terms of elementary functions and/or integrals and/or number theoretical functions would amaze everyone because it probably does not exist.

Hence the next expression we logically look for is also an iteration , just like most expressions concerning dynamical systems and tetration.

For instance (and by lack of replies of other solutions )

Let g(x) = f(x) + x = exp^(1/2)[ln^(1/2)[x]+1]

Then we state g(x) is O( x th iteration of h(x) ) where O is big O notation.

( by lack of good results I am forced to use big O at the moment )

Let g(0) = h(0) = 2.

Although I was not able to find a good h function , I considered another function called H(x).

the point of h and H is that they must have simple forms but nonconventional superfunctions ; nonconvential growth rates.

( see beginning of this post )

The H(x) I might have found ( needs confirmation and proofs etc ) comes as a special case of a generalization of something I call " a transition ".

Consider H(x,a,b) = x(1+1/ln(x)^(a+(b/x)))

For 1=<a=<5 The superfunction of H(x,a,b) is O(exp(U x^I + O)) where U,I,O depend mainly on a.

For a>=7 however the superfunction of H(x,a,b) is O(J B^x + L) where J,B,L mainly depend on a.

However near a = 5.95 it appears the superfunction behaves somewhere in between and the value of b matters more.

This is " the transition " and for a near 5.95 it is unclear if we suddenly go from O(exp(U x^I + O)) towards O(J B^x + L) or there is growth rate inbetween.

I must say I did not investigate the superfunction of this yet , nor its fractals. MAYBE WE NEED TO INVESTIGATE THE FIXPOINT AT INFINITY ??

Also I do not fully understand these " transitions " yet , e.g. are there functions with multiple transistions ?

Let H(x) = x(1+1/ln(x)^(a*+(b*/x))) where a* is about 5.95 and b* is 3.1415.

Then this H(x) might be a function that lies between polynomials and exponentiations in a nontrivial ( not asymptotic to an elementary function ) way.

Thus H(x) might be a first step to h(x).

And this might be the road to an answer.

However like I said , alot of mights and maybe's and guesses.

Hoping for progress.

Regards

Tommy1729

There might be a connection to lambert W function but that is pretty complicated and hard to explain ... or plain wrong.

It might be a mistake. further numerical , statistical and theoretical ideas are not supportive.

But maybe with other functions ...

regards

tommy1729